1D Continuous Time Fourier Transform

Basis functions (complex sinusoidal or exponential)

ϕ(x,u)=ej2πux,u∈(−∞,+∞)\phi(x,u)=e^{j2\pi ux}, u \in (-\infty,+\infty)

uu = frequency = # cycles per unit of xx, ω=2πu\omega=2\pi u (radian freq.)

Inverse transform

f(x)=F−1{F(u)}=∫−∞∞F(u)ej2πuxduf(x)=F^{-1}\{F(u)\}=\int_{-\infty}^\infty F(u)e^{j2\pi ux}\, du

Forward Transform: F(u)=⟨f(x),ϕ(x,u)⟩F(u) =\langle f(x), \phi(x,u) \rangle

F(u)=F{f(x)}=∫−∞∞f(x)e−j2πuxdxF(u)=F\{f(x)\} = \int_{-\infty}^\infty f(x) e^{-j2\pi u x} \, dx

Here we use frequency (rather then radian frequency) to define FT. Inverse transform does not need the factor of 1/2π1/2\pi

Representation of FT

#incomplete

References

  1. https://mathworld.wolfram.com/FourierTransform.html